Math 1320-9 Notes of 3/18/20

Chapter 11: Partial Derivatives

• basic topic: functions of several variables, their (partial)derivatives, and the use of those derivatives.

• Before diving into details: Here is an example of the cruxof the matter:

Supposef(x, y) = x2 sin(x+ y)

Then

• fx =

• fy =

• fxx =

• fxy =

• fyx =

• fyy =

• So we need to talk about functions of several variables,their limits, continuity, and then derivatives ...

Math 1320-9 Notes of 3/18/20 page 1

2

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11.1 Functions of Several Variables

• Defined much like function of one variable, usually in termsof a mathematical expression, although it might also bedescribed verbally, numerically by a table of values, orvisually by a surface or a set of contour lines.

• We’ll focus on the algebraic form.

• Example 3:

f(x, y) =!

9! x2! y2

Math 1320-9 Notes of 3/18/20 page 2

Graphy of 2

is the set of Domain XM such that 9 H y o

all points Cx 1,4in 423 that satisfy 93

2 42ry rthe equation

2 frT2z2q X2 42 32 42 122 9 270upper half sphere

Range 0,3

Graph

12 with(plots);3 plotsetup(ps,plotoutput=‘45.ps‘,plotoptions=‘portrait,noborder,height=500,width=500‘);4 contourplot(sqrt(9-x**2-y**2),x=-3.1..3.1,y=-3.1..3.1,5 contours=[0.0,0.25,0.5,0.75, 1.0,1.25,1.5,1.75,2.0,2.25,2.5,2.75,3.0],thick-ness=3);

678 plotsetup(gif,plotoutput=‘45a.gif‘,plotoptions=‘portrait,noborder,height=500,width=500‘);9 plot3d(sqrt(9-x**2-y**2),x=-3..3,y=-3..3);

–3

–2

–1

1

2

3

y

–3 –2 –1 1 2 3

x

Figure 1. Contour lines for f(x, y) =!

9! x2! y2.

Math 1320-9 Notes of 3/18/20 page 3

Maple

Figure 2. Surface Drawing for f(x, y) =!

9! x2! y2.

• Some interactive drawings:

• f(x, y) = xy

• f(x, y) = sin(x+ y)

• f(x, y) = e!(x2+y2)

Math 1320-9 Notes of 3/18/20 page 4

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L

Terminology

•

z = f(x, y)

• x, y are the independent variables.

• z is the dependent variable.

• The domain is the set of all points (x, y) where f can beevaluated.

• Natural Domain as before.

• The range is the set of all outputs

• Domains and Ranges of

f(x, y) =!

9! x2! y2

f(x, y) = ln(y ! x2)

Math 1320-9 Notes of 3/18/20 page 5

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Range 0,3

Domain y xµ

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Range C 00,00

Functions of Three Variables

• Same sort of ideas, except we have three independent vari-ables.

• What happens to the contour lines?

• Example:f(x, y, z) = x2 + y2 + z2.

Math 1320-9 Notes of 3/18/20 page 6

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11.2 Limits and Continuity

• Recall that in one variable we considered limits like

limx!"c

f(x) = L (1)

and also one-sided limits like

limx!"c+

f(x) = L and limx!"c!

f(x) = L (2)

• The limit (1) exists if and only if the two one-sided limits(2) exist and are equal.

• In several variables we have infinitely many, rather thanjust two, directions in which we can approach a point.

• Example:

lim(x,y)!"(0,0)

sin(x2 + y2)

x2 + y2

Math 1320-9 Notes of 3/18/20 page 7

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bing.oi o ezign.si

lim(x,y)!"(0,0)

1

x2 + y2

Math 1320-9 Notes of 3/18/20 page 8

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• Example 1:

lim(x,y)!"(0,0)

x2! y2

x2 + y2

Math 1320-9 Notes of 3/18/20 page 9

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• Example 2:

f(x, y) =xy

x2 + y2

Math 1320-9 Notes of 3/18/20 page 10

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y Mx

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• Example 3:

f(x, y) =xy2

x2 + y4

Math 1320-9 Notes of 3/18/20 page 11

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m2 isline70 x m 4 4

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